Calculus of variations in \(L^ \infty\) (Q678104)

In this paper the authors consider the Calculus of Variations problem in \(L^\infty\) in which the functional is of the form \[ F(\xi)= \underset{0<s<T}{\text{ess sup}} h\bigl(x,\xi(s),\xi'(s)\bigr). \] The authors show that the greatest quasiconvex minorant of \(F\) is the same as the lower semicontinuous regularization of the functional. The relaxed functional is lower semicontinuous in a appropriate topology and yields the existence of a minimizer. Moreover, a relaxation theorem is established, under broad assumptions on \(h\).